Free Vibration Analysis of Kirchoff Plates with Damaged Boundaries by the Chebyshev Collocation Method Eric A. Butcher and Ma en Sari Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces NM 8800, USA eab@nmsu.edu, maen@mnsu.edu Keywords: Damaged Boundaries, Kirchoff plates, Chebyshev collocation method. Abstract. This paper presents a new numerical technique for the free vibration analysis of slender Kirchoff plates with both mixed and damaged boundaries. For this purpose, the Chebyshev collocation method is applied to obtain the natural frequencies of Kirchoff plates with damaged clamped and simply supported boundary conditions. The damaged boundaries are represented by distributed translational and torsional springs. The boundary conditions are applied to the boundary points and their adjacent interior points, and are coupled with the governing equation to obtain the eigenvalue problem. Convergence studies are carried out to determine the sufficient number of grid points used. First, the results obtained for the undamaged plate with mixed boundary conditions are verified with previous results in the literature. Subsequently, the results obtained for the damaged Kirchoff plate indicate the behavior of the natural vibration frequencies with respect to the severity of the damaged boundary. Specifically, the change in the frequencies can be an indicator of the amount of boundary damage. This analysis can lead to an efficient technique for structural health monitoring of structures in which joint or boundary damage plays a significant role in the dynamic characteristics. Introduction Vibration analysis of slender plates has an important role in applications of mechanical, aerospace and civil engineering. Studying the free vibration of slender plates has had a wide range of interest in the literature. For the Kirchoff plate, the ratio of the plate thickness to edge lengths is relatively small, and thus the transverse shear and rotary inertia are neglected. The well known paper by Leissa [1] is considered as the main reference for studying the free vibration of rectangular plates, in which exact characteristic equations are obtained when two opposite edges are simply supported, and the Ritz method is applied to the remaining cases. Dickinson and Li [] the Rayleigh Ritz method with simply supported plate functions to solve the free vibration
problem. Cupial [] and Bhat [] used the Rayleigh Ritz method with orthogonal polynomials to study the free vibration of composite and isotropic plates respectively. Li and Daniels [5] applied a Fourier series method to solve the problem of free vibration of elastically restrained plates loaded with springs and masses. Li et. al [6] applied two-dimensional Fourier series supplemented with several one-dimensional Fourier series to solve the free vibration problem of plates with general elastic boundary supports. Shu and Du [7] applied the differential quadrature method to solve the free vibration problem. Bert et. al [8] applied the differential quadrature method with the δ-technique to solve the static and free vibration problems of anisotropic plates. The boundaries of a structure are usually represented in idealized form, however they are rarely ideal. In this study, the effects of the non-ideal boundary conditions (which represent damage) on the natural frequencies are studied. The same problem with a uniform cross section beam was studied by Butcher et. al [9]. Also Butcher et. al [10] studied the effect of a damaged boundary (represented by a linear spring only) for a uniform bar. Sari and Butcher [11] studied the effect of a damaged boundary for a tapered beam. Since damage causes change in the stiffness of the structure, it can be presented by a spring with a finite stiffness. The amount of damage at the boundary is represented by the tensor of boundary integrity loss first defined by Butcher [10]. In the present study the boundary conditions are coupled with the governing equation to obtain the eigenvalue problem. The boundary conditions are applied to the boundary points and their adjacent interior points. The Chebyshev collocation method has been applied to different applications due to its high rate of convergence and accuracy. Yagci et.al [1] used a spectral Chebyshev technique for solving linear and nonlinear beam equations, and the method was applied for Euler-Bernoulli and Timoshenko beams, in which the spectral-chebychev technique incorporates the boundary conditions into the derivation, thereby enabling the utilization of the solution for any linear boundary conditions without re-derivation. Furthermore, Lin and Jen [1] applied the collocation method on the laminated anisotropic plate in which the solution of the problem is assumed to be a set of Chebyshev polynomials with unknown constants. A set of collocation points, also called Gauss-Lobatto points, are selected to be substituted into those polynomials. Also, Lin and Jen [1] applied the collocation method on non-rectangular anisotropic plates, and the results were verified by comparing them with the finite element method. Moreover, Deshmukh [15] applied the method with quadratic optimization for parameter estimation for nonlinear time-varying systems, and the results were accurate. Trefethen [16] obtained the natural frequencies of a clamped square Kirchoff's plate by defining a function that satisfies the boundary conditions in the x and y directions. Luo [17] used the Chebyshev collocation method to solve a wave propagation problem in a three-dimensional cube with all sides constrained with homogenous Dirichlet boundaries, by stacking two-dimensional (x,y) slice matrices along the z direction. Ehrenstien and Peyret [18] applied the Chebyshev collocation method for solving the unsteady two-dimensional Navier-Stokes equations in vorticity-streamfunction variables. Furthermore, the method was applied to a double diffusive convection problem concerning the stability of a fluid stratified by salinity and heated from below, and the results showed excellent agreement when compared to other results available in the literature. Deshmukh et.al [19] applied the Chebyshev spectral collocation method to reduce
the dimensions of nonlinear delay differential equations with periodic coefficients. To the author s knowledge, the use of Chebyshev collocation matrices ( as opposed to Chebyshev polynomials) for studying the free vibration problem of a continuous system with higher order or mixed boundary conditions has not been done. Chebyshev Spectral Collocation Gauss-Chebyshev-Lobatto or Chebyshev extreme points are the points in the interval [-1,1] defined by x j = cos π ( j / N ), j = 0,1,, N. (1) Chebyshev points are the projections on [-1, 1] of equally spaced points on the upper half of the unit circle, and they are numbered from right to left [16]. For the set of N Chebyshev points we have an N+1 x N+1 Chebyshev differentiation matrix D N. The Chebyshev differentiation matrix is obtained by interpolating a Lagrange polynomial of degree N at each Chebyshev point, differentiating the polynomial, and then finding the derivative of the polynomial at each Chebyshev point. The entries of this matrix are N + 1 N + 1 x j DN =, D,, 1,, 1 00 N = D NN N = j = N jj 6 6 1 ( ) ( ) ( ) ( ) ( 1) i+ j ( ) ( x j ) c, 0, i i = or N DN =, i j, i, j = 0,, N. c ij i = c x x 1, otherwise. j i j () When solving ODEs or PDEs by the Chebyshev collocation method, the first derivative is represented by D1=D N, and the second derivative by D= (D1) =(D N ), and so on. The Discretized Kirchoff Plate The governing equation for a thin rectangular plate is D w + ρ w t = 0 () where D =Eh /1(1-ν ) is the flexural rigidity, = ( = / x + / y ) is the biharmonic operator, E is Young s modulus, h is the plate thickness, and ν is Poisson s ratio. iωt Assuming a harmonic solution as w( x, t) = w( x) e, the non-dimensional governing equation is given by W W + λ + λ = Ω W Y W ()
where W is the dimensionless mode shape function; Ω = ω a ρ / D is the dimensionless frequency, X=x/a, Y=y/b are the dimensionless coordinates, a and b are the lengths of the plate edges, and λ =a/b is the aspect ratio. For the Kirchoff plate, at each point on the boundaries two conditions should be satisfied, and since there is only one unknown at each point (W(i,j)), then only one boundary condition can be satisfied. In order to overcome that difficulty, the other boundary condition is applied at the adjacent interior points. For example a clamped boundary condition at the edge x=0 has the conditions W N 1, j k = 1 = 0 D1 1, k W k, j = 0 (5a) (5b) Then Eq. (5a) is applied at i=1 and j=1,,n, while Eq. (5b) is applied at i=, and j=,,,n-1. The interior points are from i=,, N-, and j=,,n-, and the other points in the domain are the boundary points. The number of the boundary points is 8N-16, and the number of the interior points is N -8N+16, where N is the number of points in the x and y directions. To apply the Chebyshev collocation method, the points are numbered in the x direction first, and then in the y direction as shown in Fig. (1) where a grid of 7 x 7 points is shown as an example. Figure (1): D 7 X 7 Chebyshev Grid Eq. () can now be discretized as ( ( D I ) + λ ( D D) + λ ( I D) ) W = Ω W
The boundary conditions are discretized as: Simply supported at X=-1 W 1j =0 (Applied at i=1, and j=1,, N) W = 0 ( D( 1,: ) I )([ ]) = 0 U B (Applied at i=, and j=,, N-1) Clamped at X=1 W Nj =0 (Applied at i=n, and j=1,, N) W = 0 (Applied at i=n-1, and j=,, N-1) ( ( D1( N,: ) I ) )([ ]) = 0 U B (6) (7) (8) (9) Free at Y=-1 W λ Y W + υ = 0 (Applied at j= 1, and i=,, N-1) ( λ ( I D( 1,: ) ) + υ( D [ 1 0 0 0] ) )([ ]) = 0 U B (10) W λ + Y ( υ) W Y = 0 (Applied at j=, and i=,, N-) ( λ ( I D( 1,: )) + ( υ) ( D D11,: ( )) )([ ]) = 0 U B (11) Other boundary conditions can be applied following the same approach. After applying the boundary conditions, the system can be written as [ BB ] [ S BI ] [ S ] [ S ] { U B} { U } { } { U I } S 0 = Ω IB II I (1) where the size of S BB is (8N-16) X (8N-16), the size of S BI is (8N-16) X (N -8N+16), the size of S IB is (N -8N+16) X (8N-16), and the size of S II is (N -8N+16) X (N -8N+16). From Eq.(1), we obtain the eigenvalue problem as
[ S IB ]{ U B } + [ S II ]{ U I } = Ω { U I } 1 ( [ S ][ S ] [ S ] + [ S ] ){ U } = Ω { U } IB BB BI II I I (1) Results for Mixed Boundary Conditions A convergence analysis was carried out to determine the minimum number of grid points to be used. Table (1) shows that N=0 collocation points is sufficient for four decimal place accuracy for the lowest four dimensionless frequencies. These frequencies were obtained for the Kirchoff plate with SS-F-SS-F and C-C-C-SS mixed boundary conditions. The results show good agreement when compared with results of Leissa [1] as seen in Tables (-). All results are for a/b =1, ν=0. and are shown to four decimal places. Table (1): Convergence analysis for a SS-F-SS-F plate N Ω 1 Ω Ω Ω 17 9.6 16.01 6.8671 9.8 18 9.61 16.15 6.710 8.9615 19 9.6 16.19 6.79 8.98 0 9.61 16.18 6.757 8.950 1 9.61 16.18 6.757 8.950 Table (): SS-F-SS-F plate Mode Leissa [1] Present Study (N=0) 1 9.61 9.61 16.18 16.18 6.756 6.757 8.950 8.950 Table () :C-C-C-SS plate Mode Leissa [1] Present Study (N=0) 1 1.89 1.86 6.7 6.1 71.08 71.078 100.8 100.79
Kirchoff Plate with Damaged Boundaries For a plate with damaged boundaries at x=a, the boundary conditions are: kw k T ( a, y) w x = D w x ( a, y) ( υ) ( a, y) ( a, y) w ( a, y) w ( a, y) = D x + + υ w x y y (1a) (1b) where k T and k are the rotational and translational springs, respectively. In terms of nondimensional parameters X=x/a and Y=y/b, Eq. [1] is written as: W ( 1, Y ) W W = α ( 1, Y ) ( υ) ( 1, Y ) ( 1, Y ) W ( 1, Y ) W ( 1, Y ) + = α t + υ W λ Y Y (15a) (15b) where α and α t are the dimensionless damage parameters defined as : D α =, α t = a k D ak t (16) Note that if the boundary is undamaged then k=k t = and α=α t =0. Similarly, if a damaged boundary is at y=0 or y=b, the damage parameters will be the same, except that a will be replaced by b. In Figs. - the first three natural frequencies are plotted versus the damage parameters α and α t for different boundary conditions. Ω 75 70 6.95 60 55 50 0.08 5 0.0 0 0 0.05 0.1 0.15 0. 0.5 0. 0.5 0. 0.5 0.5 Figure (): The first three natural frequencies versus α =α t for a D-C-C-C square plate. α=α t
70 60 Ω 1.7017.069 0 1.6871 10 0 0.05 0.1 0.15 0. 0.5 0. 0.5 0. 0.5 0.5 α=α t Figure (): The first three natural frequencies versus α = α t for a SS-C-SS-D square plate. 70 65 60 55 50 Ω 6.9 0 16.1 9.61 0 0.05 0.1 0.15 0. 0.5 0. 0.5 0. 0.5 0.5 α=α t Figure (): The first three natural frequencies versus α = α t for a SS-D- SS-D square plate. It is observed in Figs.(-) that when the damage parameter vanishes, which means that no damage is introduced at the associated boundary, the frequencies correspond to that of a clamped
boundary. On the other hand, as the damage parameter increases the natural frequencies decrease, and if the parameter of damage equals infinity (totally damaged), the frequencies correspond to those of a free boundary (these values are shown by the dashed lines in Figs. (- )). Furthermore, in Fig.() the eigenvalue veering phenomenon is observed between the second and third modes at α=α t 0.05. Eigenvalue veering refers to a region in which two eigenvalue loci approach each other and almost cross as the system parameter is changed, but instead of crossing they appear to veer away from each other [1]. The second and third mode shapes are plotted for α=α t =0.0 and 0.0 before and after the veering phenomenon is observed, respectively, as shown in Fig.(5). Moreover, in Fig.() there is also veering between the second and third mode shapes at α=α t 0.01, and these mode shapes are generated for α=α t =0.005 and 0.015, respectively, as shown in Fig.(6). These results clearly show an exchange of mode shapes takes place during the veering as the damage parameter is increased. α=α t =0.0 Ω : Ω : α=α t =0.0 Ω : Ω : Figure (5): The second and third mode shapes before and after the veering of the SS-C-SS-D square plate in Fig.()
α=α t =0.005 Ω : Ω : α=α t =0.015 Ω : Ω : Figure (6): The second and third mode shapes before and after the veering of the SS-D-SS-D square plate in Fig.(). Conclusions A new approach for the free vibration analysis of slender Kirchoff plates with damaged boundaries was investigated, where the Chebyshev collocation method was applied to solve the eigenvalue problem. Convergence analysis was carried out to determine the sufficient number of points to use in x and y directions, and the boundary conditions were coupled with the governing equation. The natural frequencies of several undamaged cases with mixed boundaries were calculated and compared with the results of Leissa [1]. For damaged boundaries, plots of the first three natural frequencies versus the damage parameters were generated, where it was shown that as the damage increases the natural frequencies decrease. Also, the veering phenomenon was observed between the second and third mode shapes for the SS-C-SS-D and the SS-D-SS-D cases, and the mode shapes were generated at values of the damage parameters before and after the veering, during which an exchange of mode shapes was demonstrated. To the authors knowledge, it is believed that this is the first time that the Chebyshev collocation method is applied to a system with higher order or mixed boundary conditions in an efficient way. The authors hope to use this method to develop an efficient technique for structural health monitoring of structures in which joint or boundary damage plays a major role in the dynamic characteristics. Current work is focused on extending the above analysis to the case of Mindlin plates with damaged boundaries. Acknowledgments Financial support of NASA under Grant No. GR00088 is gratefully acknowledged.
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